Asymptotic Enumeration of Hypergraphs by Degree Sequence

We prove an asymptotic formula for the number of k-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of d-regular k-uniform hypergraphs on n vertices provided that dn ≤ c(n/k) for a constant c > 0, and 3 ≤ k < n^c for any C < 1/9. Our results relate the degree sequence of a random k-uniform hypergraph to a simple model of nearly independent binomial random variables, thus extending the recent results for graphs due to the second and third author.